Artificial earth satellites offer a unique opportunity to exploit the possibility of deducing temperature profiles on a global scale from measurements of radiance in several narrow spectral intervals in a strongly absorbing band of an atmospheric gas whose mixture is uniform. In the earth's atmosphere the 4.3-micron and 15-micron bands of carbon dioxide and the 5-mm. band of oxygen may be used; only the 15-micron band is considered in detail, although the procedures are applicable to the other bands. The problem considered is the numerical solution of the integral form of the radiative transfer equation from measurements in a finite set of spectral intervals. It is shown that, by a suitable approximation of the Planck radiance, the radiative transfer equation can be reduced to an integral equation of the first kind. After a discussion of the kernel, which is associated with the transmittance of the gas, the equation is changed to a finite set of equations which is amenable to numerical solution. The solution is limited to about six pieces of information, which may be expressed as points along the vertical profile, or as coefficients of an expansion; the limitation in information is manifest in the transmittance curves for the several spectral intervals, the errors of measurement, and the approximations employed. However, even in this limited case the formal solution of the set of equations is unstable. A method of stabilizing the solution by smoothing is discussed. In this process the amount of smoothing remains small, so that the inherent properties of the temperature profile are not affected. Several possible forms of the solution are discussed, and it is concluded that empirical orthogonal functions are preferred because they contain the physical information lacking in analytical forms. Examples are shown of solutions for radically different profiles, both with “exact” simulated measurements and with random errors introduced.